Semiprime CS group algebra of polycyclic-by-finite group without domains as summands is hereditary

Behn showed that if K[G] is a prime group algebra with G polycyclic-by-finite, then K[G] is a CS-ring if and only if K[G] is a pp-ring if and only if G is torsion-free or G≅D∞ and char(K)≠2. As a consequence, such a group algebra K[G] is hereditary excepting possibly when K[G] is a domain. In this paper we show that if K[G] is a semiprime group algebra of polycyclic-by-finite group G and if K[G] has no direct summands that are domains, then K[G] is a CS-ring if and only if K[G] is hereditary if and only if G/Δ+(G)≅D∞ and char(K)≠2. Precise structure of a semiprime CS group algebra K[G] of polycyclic-by-finite group G, when K is algebraically closed, is also provided.